IJESAT_2012_02_03_05

ANUPAM TIWARI ISSN: 22503676

[IJESAT] INTERNATIONAL JOURNAL OF ENGINEERING SCIENCE & ADVANCED TECHNOLOGY Volume-2, Issue-3, 406 411

IJESAT | May-Jun 2012

Available online @ http://www.ijesat.org 406

FRACTAL APPLICATIONS IN ELECTRICAL AND ELECTRONICS

ENGINEEERING

Anupam Tiwari

Member (L-209735), Institute of Electronics and Telecommunication Engineers N Delhi, India

[email protected]

Abstract Fractal, which means broken or irregular fragments, was originally coined by Dr B. Mandelbrot in the year the 1970. Fractals

describe a family of complex shapes that possess an inherent self-similarity in their geometrical structure usually in nature, since the

pioneering work of Mandelbrot and others, a wide variety of applications for fractals has been found in many branches of science and

engineering. A fractal is a recursively generated object having a fractional dimension. Fractals are defined as the set whose

Hausdorff dimension exceeds topological dimension. Fractal has various properties like recursive, infinite, self- symmetry and

fractional dimension (FD). Self Symmetry, Space filling and FD are most important properties which has practical applications. This

paper discuss various applications of fractal in electrical and electronics engineering field few are fractal antenna, fractal capacitor,

fractal image compression, fractal encoding, fractal analysis in power systems fault analysis like (High Impedance fault, Load

forecast). Hurst parameter is used for real dynamic curve analysis. Lightning modeling is essential for design and testing of an

aircraft in laboratory. Full scale aircraft testing of lightning stroke inside lab fractal lightning modeling is used to generate High

Volt lighting discharge testing of full model aircraft on ground. Impedance matching by impedance transformer by fractals is new

concept for matching loads in electronics. L-System is extensively used for biological modeling and growth modeling. L-System is also

used for artificial life creation by computers. L-System fractals can be used for digital camouflaged, concealed super structures,

computer generated tree and artificial life. Scale free networks are fractal networks, they are more robust. Hurst parameter (H), of

traffic for scale free networks if change abruptly, it signifies attack on network. It is also proposed to monitor H for scale free

network. Routing protocol and optimization techniques in networks should also choose nodes so that the network grows and forms like

fractal network which are robust.

Index Terms: FD, Correlation dimension, Self-symmetry, Space-filling, Hurst Parameter (H), L-System, fractal Radio,etc

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1. INTRODUCTION

Our world and its real world problems are highly nonlinear

and fractal, which cannot be described in terms of Euclidian

geometry. Fractal geometry is considered fashionable

mathematics, humanistic and quasi pragmatic. Besides, its

connection with nature, art, science, engineering, the marvels

of computer and IT development. Fractals geometry and

modeling is present in almost all the natural creation. It has

also impact on a special nature beauty, and a particular way of

thinking, that can be felt in the mind and heart.

Fractal has important properties, like FD, self symmetry and

space filling, which can be used for interpretation of real

world nonlinear problems and systems. The dimension of the

geometry can be interpreted as a quantification of the space

filling ability of the geometry. While Euclidean geometries

have integer dimensions, for example dimension of line is 1,

and 2 for a plane, but fractals have fractional dimensions.

Space filling curves after infinite iteration may achieve integer

dimensions. Combining fractal geometry with dynamical

systems lead to non-linear dynamical systems, which can help

in solving problems in our current technological era that

Einsteins theory, was unable to solve problem like power

system dynamics. Fig-1 shows iterations of Sierpinski fractals

and Fig-2, various types of Fractals.

Fig-1: Iteration of Serpinski Triangle

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Fig-2: Types of Fractals

2. MATHEMATICAL BACKGROUND

FD can be defined in terms of various methods depending

upon type of system become fractal. Box dimension method is

usually for deterministic systems like measuring of coast line,

Correlation dimension is for random system like measurement

of dimension of strange attractor of dynamical system and for

time series analysis H which is related to FD.

Hausdorff dimension, how does the number of balls it takes to cover the fractal scale with the size and

radius of the ball.

Box-counting dimension, how does the number of boxes it takes to cover the fractal scale with the size

of the boxes.

Information dimension, how does the average information needed to identify an occupied box scale.

Correlation dimension, calculated from the number of points used to generate the picture, and the number of

pairs of points within a distance of each other.

If s is the scale factor and Hausdorff dimension of an

object is based on covering the object by small disks or balls

for a minimum cover. The box dimension of a subset X of the

plane is defined similarly by counting the number of cells of a

grid with constant, s that intersect X, N(s). Then X has a

dimension D if, N (s) satisfies the power law.

N(s) = c(1/s)D (1)

Asymptotically in the sense that

lim N(s) sD

= c. (2)

s 0

The box dimension D can be computed from (2) as

D = lim [- log N(s)/log(s)]. (3)

s 0

Fractal dimension as defined by Equation (3) is usually

identified with HausdorffBesicovitch dimension and is

known as the capacity dimension. There is a fine distinction

between the HausdorffBesicovitch dimension and the

capacity dimension while the former is obtained by covering

the set minimally with hyper cubes that may be different in

size; the latter is obtained with the same process except that

the hyper cubes are the same size. Spatial correlation is

measured in terms of correlation dimension.

H is Heaviside function.

Hurst parameter (H) is related with roughness of dynamic time

series curve like Load curve of power plant, and can be used

as a measure of FD. Estimating the Hurst exponent for a data

set provides a measure of whether the data is a pure random

walk or has underlying trends All the known representations

of such bursts in processes have one common parameter value.

This parameter has different but equivalent representations

and names. That is H, FD, and the exponent of the 1/ f

process power spectrum are related as

= 2H 1 = 5 2D

There is also a form of self-similarity called statistical self-

similarity. Assuming that we had one of those imaginary

infinite self-similar data sets, any section of the data set would

have the same statistical properties as any other. Statistical

self-similarity occurs in a surprising number of areas in

engineering, computer network traffic traces are self-similar.

Other examples of statistical self-similarity exist in

cartography (the measurement of coast lines), computer

graphics (the simulation of mountains and hills), biology

(measurement of the boundary of a mould colony) and

medicine (measurement of neuronal growth). Self-symmetry

means superstructure is formed from small structure which is

having scale invariance.

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3. APPLICATIONS IN ELECTRICAL AND

ELECTRONICS ENGINEERING

There are various applications of fractals in electrical and

electronics engineering field some are:

3.1 High Impedance Fault Detection

In Power Systems, high impedance faults are very peculiar,

these faults dont draw enough current to operate protective

relays thus very difficult to predict. Phase currents and voltages in a distribution power system change with a certain

degree of chaos when high impedance faults (HIFs) occur

concepts of fractal geometry to analyze chaotic properties of

high impedance faults. These faults also fail to establish return

path, as fault progress it melts conductor, causes arcing,

displacement of soil etc causes current very chaotic. FD is

measure of this fault as shown in Fig3.

Fig-3: Block Diagram of Fault Locator

3.2 Fractal Capacitor

Unlike conventional metal-to metal capacitors, the density of a

fractal capacitor increases with scaling. The capacitance per

unit area of a fractal structure depends on the dimension of the

fractal. To improve the density of the layout, fractals with

large dimensions should be used. Fractals add one more

degree of freedom to the design of capacitors, allowing the

capacitance density to be traded for a lower series resistance.

Most of the fractal geometries randomize the direction of the

current flow and thus reduce the effective series inductance;

fractals usually have lots of rough edges that accumulate

electrostatic energy more efficiently compared to inter

digitized capacitors.

3.3 Lightning Modelling

Lightning modelling is realised with fractal. Lightening

modelling is used various applications. Study of lightning

stroke on full scale model of aircraft on ground requires

lightning stroke which is generated by fractal modelling. High

voltage equipment full scale testing, and dielectric breakdown

studies uses FD as a parameter.

3.4 Load Scheduling

Load curve is highly dynamic and depending on of various

random parameters like, temperature, humidity, time of the

day and social parameters like festivals etc. Roughness of

curve can be linked with FD. Roughness as measure can be

used by Artificial Intelligence (AI) systems for load

scheduling.

3.5 Fractal Antenna Multiband/ Miniaturization

The next major opportunity for wireless component

manufacturers is to put the antenna into the package [1].

Fractal-technology applied to antennas reaches the required

miniaturization to make Full Wireless System in Package

(FWSiP) a reality [2].The space filling properties of fractals

enable the production of miniature antennas with optimal

performance, and with multi band capabilities. Fractal

Antenna technology is suitable for Bluetooth, WLAN, GPS,

UWB and Zigbee and for sensors for automotive, biomedical

and industrial purposes. Some key benefits of fractals in

antenna geometry are:

Broadband and multiband frequency response that derives from the inherent properties of the fractal

geometry of the antenna.

Compact size compared to antennas of conventional designs (miniaturization), while maintaining good to

excellent efficiencies and gains.

Mechanical simplicity (no matching) and robustness. Characteristics of the fractal antenna are obtained due

to its geometry and not by the addition of discrete

components.

Design to particular multi-frequency characteristics containing specified stop bands as well as specific

multiple pass bands.

Apollonian gasket fractal antenna with CPW (Co-planer

waveguide) monopole feed shown in Fig-4. The antenna is

designed and fabricated on FR4 material with dielectric

constant (4.33). Experimental result in laboratory condition

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measured return loss (S11 -10 dB parameter) by Vector

analyzer shows the multiband behavior with the centre

frequencies of 1.265 GHz, 4.66 GHz and 7.8 GHz with

bandwidth of 50%, 17.5% and 15 % respectively. This

multiband is achieved with fractal geometry [3].

Fig-4: Apollonian Gasket CPW monopole Fractal Antenna

.

3.6 Fractal Encoding in Communication

Fractal-coded images are decompressed with a low-

complexity decoding scheme [4]. The fractal-based

compression utilizes local self similarity in images of textures

and natural scenes. Fractal image textures offer significant

savings in both storage and transmission bandwidth. Fractal

compression/decompression could surpass the JPEG standard

in mobile applications as shown in Fig-5.

Fig-5: Fractal Coding

3.7 Fractal Image Compression

The algorithm involved in compressing an image is

Specifying the rate (bits available) and distortion (tolerable error) parameters for the target image.

Dividing the image data into various classes, based on their importance.

Dividing the available bit budget among these classes, such that the distortion is a minimum.

Quantize each class separately using the bit allocation information derived in step 3.

Encode each class separately using an entropy coder and write to the file.

3.8 Impedance Transformer

Using a Sierpinski fractal micro-strip stepped-impedance

transformers using a Sierpinski fractal shape are proposed for

the first time as shown in Fig-6. The proposed fractal- shaped

stepped-impedance transformers can greatly enhance the

operating frequency bandwidth. Fractal Shape has enhanced

bandwidth up to 44.0% and 73.9% (S11-30 dB) has been

achieved by the presented two-section and three-section

fractal-shaped maximal flat stepped-impedance transformers

[5]. Z1, Z2, Z3 are Impedances, W1, W2, W3 are Width and

g1, g2, g3 are wavelengths where, l1= , l2= ,

l3= .

Fig-6 Fractal Impedance Transformer for matching load





3.9 Artificial Life (Computer Graphics)

Fractals are commonly used in computer graphics, not only for

creating visual effects, such as fires, clouds, and lightning, but

also for modeling living organisms Lindenmayer system (L-

system) is a formal grammar (a set of rules and symbols) most

famously used to model the growth processes of artificial plant

development as shown in Fig-7, it is used to model the

morphology of a variety of organisms [6]. Furthermore, the

interest of biologists in modeling actual plant species is

complemented by the fundamental studies of emergence in the

field of artificial life. These varied interests and applications

place L-systems in the center of interdisciplinary studies

bridging theoretical computer science, computer graphics,

biology and artificial life. Commands for Turtle action are as

shown in Table-1.

Symbol Meaning

F Draw forward by a fixed length.

G Go forward without drawing a line.

+ Turn right by a fixed angle, n+ means turn n times.

- Turn left by a fixed angle, n- means turn n times.

[ Push (put) current position to stack.

] Pop (take) current position from stack.

|

Draw forward by a length that depends on the

execution depth.

Table-1: Commands for Turtle Grammar

Fig-7 Fractal L-System (Artificial Plant)

Fibonacci sequence is used by most of the biological modeling

by L-System as shown in Table-2.

Initial state: A: B

B: AB

And so on.1 1 2 3 5 8 13 ........

n Series

n=0 A

n=1 AB

n=3 BAB

n=4 ABBAB

n=5 BABABBAB

Table-2: Fibonacci Series

3.10 Fractal Radio and Fractal Radar Concept

Application of fractal theory in combination with the theory of

fractional integrodifferential operators and fractal

interpretation to various problems of modern radio

engineering is widely accepted Fractal methods used to

process very weak signals and low-contrast images on the

basis of application of non-Gaussian statistics, to measure

new attributes of signals scattered by a statistically rough

surface, and to consider scaling effects of real radio signals

and electromagnetic fields can be used for the development of

new efficient fractal electronic systems with new capabilities,

which are sometimes unreachable for the existing facilities

designed on the basis of principles of traditional electronics.

In Fractal Radar, tone and texture and structure features on the

radar imagery are important bases of target recognition, FD is

used to extract the texture and structural features of ground

object.

3.11 Scale Free Networks

Social behaviour and spread of internet as an outcome of

social behaviour and communications leads to scale free

networks called Francnet [7]. A scale-free network is a

network whose degree distribution follows a power law, at

least asymptotically. That is, the fraction P (k) of nodes in the

network having k connections to other nodes goes for large

values of k as

P (k) ~ ck-

Where c is a normalization constant and is a parameter

whose value is typically in the range 2 < < 3, at times it may

lie outside these bounds. Scale free networks are very robust

to errors be optimized for fractal networks. attacks compared

to scale-free networks with the same exponent[8] .Thus, it

seems that fractality provides better protection when the hubs

are removed from the system .This can be attributed to the

isolation of the hubs from each other and can provide an

explanation on why most biological.

http://www.mathdaily.com/lessons/Formal_grammarhttp://www.mathdaily.com/lessons/Planthttp://en.wikipedia.org/wiki/Complex_networkhttp://en.wikipedia.org/wiki/Degree_distributionhttp://en.wikipedia.org/wiki/Power_law





Scale free networks are very sensitive to targeted attacks, so it is proposed to monitor H, in scale free

network traffic flow for detecting attacks. Routing

algorithm also select path nodes as per scale free

fractal networks for robustness of network.

4. CONCLUSION

Nature adapts Fractal modelling for its creation. Soft

computing incorporates fractal properties like fractal

dimension, self symmetric, scale invariance in describing and

modelling physical phenomenon. Fractal parameters and

properties like Space filling, FD, H and self symmetric are

used for various analyses. L-system is used for biological

modelling. Fractal radio and Fractal Radar concepts are new

and requires deep research. Fractal concepts on social

networking, Scale free networks and internet are also utilised

for robust and fault analysis. Protocols and security system can

be designed with fractal parameters. Routing protocol should

be designed to select node based on scale free network for

better stability of network. Security and network monitoring

system can detect an attack by monitoring the networks Hurst

parameter. It is proposed to monitor Hurst parameters for scale

free network for detection of attack.

REFERENCES

[1] Nathan Cohen, Fractal antenna Part1 and Part2.

[2] S Wong , B L Ooi Analysis and bandwidth

Enhancement of Sierpenski carpet Antenna C. Puente, J.

Romeu, R. Pous, X. Garcia and F. Benitez Fractal multiband

antenna based on the Sierpinski gasket

[3] Rajkumar, AnupamTiwari On Design of CPW fed

Apollonian Gasket multiband antenna International Journal

on microwave and Optical Letter. Vol-4, No-3 May 2009.

[4] Shicai Lin, Tiecai Li and Xianglong Tang Achieving

Low-Power Consumption by Using Fractal Coding in PVP

Proceedings of the 2007 IEEE International Conference on

Mechatronics and Automation

[5] Wen-Ling Chen, Guang-Ming Wang,Yi-Na Qiand Jian-

Gang Liang , fractal-shaped stepped impedance transformers

for wideband application, Microwave and Optical

Technology Letters / Vol. 49, No. 7, July 2007

[6] P furmanek Polyhedral based covers based on L-system

construction The journal of Polish society of Geometry and

engineering Graphics, vol 14, 2004, pp40-47.

[7] K.Park and W.Willinger, Eds., Self-Similar Network

Traffic and Performance Evaluation. Wiley Interscience, 2000.

[8] A. Adas, Traffic Models in Broadband Networks, IEEE

Communications Magazine, Jul. 1997. p.82-89.

BIOGRAPHIES

Anupam Tiwari was born in Kanpur,

U.P. on 9th July 1972, received M Tech

(Modeling and Simulation) from Defence

Institute of Advanced Technology-(DU),

Pune. He had also received BE (Hons) in

Electrical Engg and Topper of the course

from Assam Engineering College,

Guwahati. He received MSc degree

(Disaster Mitigation), PG diploma in Thermal Power Plants

and a certified Power Engineer from CEA. He is a

Radiological Safety Officer Level-1 from BARC and Life

Member of IETE. He has 19 papers in Journals, and

International/National conferences. His field of interest is

Modeling and Simulation, Antennas design, Wireless

communications and Radar technology.

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IJESAT_2012_02_03_05